Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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UNIFORM DISTRIBUTIONS ON CURVES AND QUANTIZATION

  • Received : 2021.12.28
  • Accepted : 2022.07.28
  • Published : 2023.04.30
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Abstract

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of n-means and the nth quantization errors for different values of n with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise [2], which says that for a Borel probability measure P with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.

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References

  1. E. F. Abaya and G. L. Wise, Some remarks on the existence of optimal quantizers, Statist. Probab. Lett. 2 (1984), no. 6, 349-351. https://doi.org/10.1016/0167-7152(84)90045-2
  2. J. A. Bucklew and G. L. Wise, Multidimensional asymptotic quantization theory with rth power distortion measures, IEEE Trans. Inform. Theory 28 (1982), no. 2, 239-247. https://doi.org/10.1109/TIT.1982.1056486
  3. C. P. Dettmann and M. K. Roychowdhury, Quantization for uniform distributions on equilateral triangles, Real Anal. Exchange 42 (2017), no. 1, 149-166. http://projecteuclid.org/euclid.rae/1490580015 https://doi.org/10.14321/realanalexch.42.1.0149
  4. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academy publishers, Boston, 1992.
  5. R. M. Gray, J. C. Kieffer, and Y. Linde, Locally optimal block quantizer design, Inform. and Control 45 (1980), no. 2, 178-198. https://doi.org/10.1016/S0019-9958(80)90313-7
  6. S. Graf and H. Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics, 1730, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/BFb0103945
  7. R. M. Gray and D. L. Neuhoff, Quantization, IEEE Trans. Inform. Theory 44 (1998), no. 6, 2325-2383. https://doi.org/10.1109/18.720541
  8. A. Gyorgy and T. Linder, On the structure of optimal entropy-constrained scalar quantizers, IEEE Trans. Inform. Theory 48 (2002), no. 2, 416-427. https://doi.org/10.1109/18.978755
  9. S. Matsuura, H. Kurata, and T. Tarpey, Optimal estimators of principal points for minimizing expected mean squared distance, J. Statist. Plann. Inference 167 (2015), 102-122. https://doi.org/10.1016/j.jspi.2015.05.005
  10. J. Rosenblatt and M. K. Roychowdhury, Optimal quantization for piecewise uniform distributions, Unif. Distrib. Theory 13 (2018), no. 2, 23-55. https://doi.org/10.2478/udt-2018-0009
  11. M. K. Roychowdhury, Optimal quantizers for some absolutely continuous probability measures, Real Anal. Exchange 43 (2018), no. 1, 105-136. https://doi.org/10.14321/realanalexch.43.1.0105
  12. R. Zam, Lattice Coding for Signals and Networks: A Structured Coding Approach to Quantization, Modulation, and Multiuser Information Theory, Cambridge University Press, 2014.